ISEE Middle Level Quantitative Reasoning Flashcards: Proportional Relationships

What this deck covers

This deck focuses on Proportional Relationships, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Middle Level Quantitative Reasoning.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the missing value $x$ in the scale ratio $ \frac{3}{5}=\frac{12}{x}$?

Answer: $x=20$. Cross-multiplying yields $3x = 60$, so dividing by 3 gives $x=20$.

Flashcard 2: What percent is $18$ of $72$ using $ \frac{\text{part}}{\text{whole}}=\frac{p}{100}$?

Answer: $25\%$. Setting up $18/72 = p/100$ and solving gives $p = (18/72) imes 100 = 25$.

Flashcard 3: What is the distance traveled in $5$ hours at a constant speed of $42$ miles per $3$ hours?

Answer: $70$ miles. The speed is $42/3 = 14$ mph, so in 5 hours the distance is $5 imes 14 = 70$ miles.

Flashcard 4: What is the cost of $15$ items if $6$ items cost \10$ (constant unit price)?

Answer: \25$. The unit price is$10/6 = 5/3$, so for 15 items the cost is$15 imes (5/3) = 25$.

Flashcard 5: What is the unit rate if $18$ items cost \24$?

Answer: \\frac{4}{3}$per item. Dividing total cost by number of items gives the rate of$24/18$, which simplifies to$4/3$ per item.

Flashcard 6: Identify whether $ \frac{5}{6}$and$ \frac{20}{24}$ are proportional.

Answer: Yes, because $5\cdot24=6\cdot20$. The cross-products are equal at 120, confirming the ratios are proportional.

Flashcard 7: Which option shows ratios that are proportional: A) $ \frac{2}{3}$and$ \frac{8}{12}$, B)$ \frac{2}{3}$and$ \frac{8}{10}$?

Answer: A) $\frac{2}{3}$ and $\frac{8}{12}$. Simplifying $8/12$ to $2/3$ shows equivalence to the first ratio, while $8/10$ simplifies to $4/5$, which differs.

Flashcard 8: What is $x$ if $y=kx$, $k=\frac{5}{3}$, and $y=25$?

Answer: $x=15$. Rearranging $y = kx$ to $x = y/k$ and substituting gives $x = 25 / (5/3) = 15$.

Flashcard 9: What is $y$ if $y=kx$, $k=\frac{3}{4}$, and $x=20$?

Answer: $y=15$. Substituting into the equation yields $y = (3/4) imes 20 = 15$.

Flashcard 10: What is $k$ if $y$ is proportional to $x$ and $(x,y)=(6,15)$?

Answer: $k=\frac{5}{2}$. Dividing $y$ by $x$ gives the constant $k=15/6$, which simplifies to $5/2$.

Flashcard 11: What is $x$ if $ \frac{3}{8}=\frac{6}{x}$?

Answer: $x=16$. Cross-multiplying results in $3x = 48$, so dividing by 3 provides $x=16$.

Flashcard 12: What is $x$ if $ \frac{9}{12}=\frac{x}{20}$?

Answer: $x=15$. Cross-multiplying yields $180 = 12x$, and dividing by 12 gives $x=15$.

Flashcard 13: What is $x$ if $ \frac{7}{x}=\frac{21}{12}$?

Answer: $x=4$. Cross-multiplying gives $84 = 21x$, so dividing by 21 solves for $x=4$.

Flashcard 14: What is $x$ if $ \frac{x}{5}=\frac{12}{15}$?

Answer: $x=4$. Cross-multiplying the proportions $15x = 60$ and dividing by 15 yields $x=4$.

Flashcard 15: What is the percent proportion used to find a part of a whole?

Answer: $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$. The percent proportion sets the ratio of part to whole equal to the percent over 100 to solve for unknowns.

Flashcard 16: Identify the proportional equation for a constant ratio $ \frac{y}{x}=k$.

Answer: $y=kx$. This equation captures the direct variation where $y$ is always a constant multiple $k$ of $x$.

Flashcard 17: What does it mean for two ratios to be proportional?

Answer: They are equal: $ \frac{a}{b}=\frac{c}{d}$with$b\ne^0$and$d\ne^0$. Two ratios are proportional when they express the same relationship, making their fractions equivalent provided the denominators are not zero.

Flashcard 18: What is the direct method to find a missing value in $ \frac{a}{b}=\frac{c}{x}$?

Answer: $x=\frac{bc}{a}$ (with $a\ne^0$). Cross-multiplication equates $a x = b c$, so isolating $x$ gives the product of $b$ and $c$ divided by $a$.

Flashcard 19: What is the direct method to find a missing value in $ \frac{a}{b}=\frac{x}{d}$?

Answer: $x=\frac{ad}{b}$ (with $b\ne^0$). Solving for the missing value uses cross-multiplication to equate products, yielding $x$ as the product of $a$ and $d$ divided by $b$.

Flashcard 20: What is the unit rate for a ratio written as $ \frac{a}{b}$?

Answer: $\frac{a}{b}$ per $1$ (the value when the second quantity is $1$). The unit rate simplifies the ratio to express the value of the first quantity when the second is exactly 1.

Flashcard 21: What point must be on the graph of any proportional relationship $y=kx$?

Answer: $(0,0)$. The graph of $y=kx$ is a straight line through the origin, so it always passes through $(0,0)$ regardless of $k$.

Flashcard 22: What is the equation of a proportional relationship using constant $k$?

Answer: $y=kx$. A proportional relationship is modeled by this linear equation where $y$ varies directly with $x$ through the constant $k$.

Flashcard 23: What is the constant of proportionality $k$ in $y=kx$?

Answer: $k=\frac{y}{x}$ for $x\ne^0$. The constant of proportionality $k$ represents the fixed ratio of $y$ to $x$ in a direct proportion, defined as their quotient when $x$ is not zero.

Flashcard 24: What is the cross-products test for $ \frac{a}{b}=\frac{c}{d}$?

Answer: Proportional if and only if $ad=bc$ (with $b\ne^0$ and $d\ne^0$). The cross-products test checks equality by verifying if the product of the numerator of one ratio and the denominator of the other equals the reverse, excluding zero denominators.

Flashcard 25: Find $y$ if $y$ is proportional to $x$ and $y=14$ when $x=8$, then $x=20$.

Answer: $y=35$. The constant $k=14/8=7/4$, so for $x=20$, $y=20 imes (7/4)=35$.